Another way to define a fair game is that a player’s probability of winning must be equal to the player’s share of the pot of money awarded to the winner. All money is put into a pot at the start of the game, and the winner claims the entire amount that is in the pot. (Compare this definition to that in Exercise 31.)
(a) The player and the host each put $20 into the pot. The player rolls a die and wins the pot if the die produces an even number. Is this game fair to the player?
(b) A player at a casino puts $1 into the pot and names a card from a standard deck (e.g., ace of diamonds). The casino puts $99 into the pot. If a randomly selected card matches the choice of the player, the player wins the pot. Is this game fair to the player?
Exercise 31
A game involving chance is said to be a fair game if the expected amount won or lost is zero. Consider the following arcade game. A player pays $1 and chooses a number from 1 to 10. A spinning wheel then randomly selects a number from 1 to 10. If the numbers match, the player wins $5. Otherwise the player loses the $1 entry fee.
(a) Define a random variable
that is the amount won by the player and draw its probability distribution. (Capital letters other than
can be used for random variables.) Use negative values for losses, and positive values for winnings.
(b) Find the mean of
. Is this a fair game?